cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A188687 Partial binomial sums of binomial(3n,n)/(2n+1) = A001764(n).

Original entry on oeis.org

1, 2, 6, 25, 126, 704, 4183, 25897, 165166, 1077520, 7156352, 48222354, 328859011, 2265428728, 15740837575, 110187356134, 776336572878, 5501042194580, 39177463572112, 280277949384146, 2013277273220064, 14514764553512488, 104993261648226446
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A188675, A188676, A104859, A188678, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]Binomial[3k,k]/(2k+1),{k,0,n}],{n,0,22}]
  • Maxima
    makelist(sum(binomial(n,k)*binomial(3*k,k)/(2*k+1),k,0,n),n,0,20);

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(3k,k)/(2k+1).
G.f.: (2/sqrt(3x*(1-x)))*sin((1/3)*arcsin(3/2*sqrt(3*x/(1-x)))).
Recurrence: 2*n*(2*n+1)*a(n) = (39*n^2-35*n+8)*a(n-1) - 2*(n-1)*(33*n-32)*a(n-2) + 31*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 31^(n+3/2)/(3^4*2^(2*n+2)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x) * A(x)^3. - Ilya Gutkovskiy, Jul 25 2021

A188678 Alternate partial sums of binomial(3*n,n)/(2*n+1).

Original entry on oeis.org

1, 0, 3, 9, 46, 227, 1201, 6551, 36712, 209963, 1220752, 7193888, 42873220, 257957352, 1564809168, 9559946496, 58768808463, 363261736872, 2256369305793, 14076552984507, 88163556913188, 554148894304557, 3494365949734563
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

Programs

  • Mathematica
    Table[Sum[Binomial[3k,k](-1)^(n-k)/(2k+1),{k,0,n}],{n,0,20}]
  • Maxima
    makelist(sum(binomial(3*k,k)*(-1)^(n-k)/(2*k+1),k,0,n),n,0,20);

Formula

a(n) = Sum_{k=0..n} binomial(3*k,k)*(-1)^(n-k)/(2*k+1).
Recurrence: 2*(2*n^2+9*n+10)*a(n+2)-(23*n^2+63*n+40)*a(n+1)-3*(9*n^2+27*n+20)*a(n)=0.
G.f.: 2*sin((1/3)*arcsin(3*sqrt(3*x)/2))/((1+x)*sqrt(3*x)).
a(n) ~ 3^(3*n+3+1/2)/(31*sqrt(Pi)*n^(3/2)*2^(2*n+2)). - Vaclav Kotesovec, Aug 06 2013
G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^2 * A(x)^3. - Ilya Gutkovskiy, Jul 25 2021

A188680 Alternate partial sums of binomial(3n,n)^2.

Original entry on oeis.org

1, 8, 217, 6839, 238186, 8779823, 335842273, 13185196127, 527732395714, 21438596184911, 881264330165314, 36575197658193086, 1530121867019096914, 64443673226319500222, 2729760145163758146178, 116203781083772019594878
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188681, A188682, A188683, A188684, A188685, A188686, A188687.
Cf. Alternate partial sums of binomial(k*n,n)^2: A228002 (k=2), this sequence (k=3).

Programs

  • Mathematica
    Table[Sum[Binomial[3k,k]^2(-1)^(n-k),{k,0,n}],{n,0,20}]
  • Maxima
    makelist(sum(binomial(3*k,k)^2*(-1)^(n-k),k,0,n),n,0,20);
  • PARI
    a(n)=my(t=1); sum(k=1,n, t*=(27*k^2 - 27*k + 6)/(4*k^2 - 2*k); (-1)^(n-k)*t^2)+(-1)^n \\ Charles R Greathouse IV, Nov 02 2016

Formula

a(n) = sum(C(3k,k)^2*(-1)^(n-k), k=0..n).
Recurrence: 4*(2*n^2+7*n+6)^2 * a(n+2) -(713*n^4+4262*n^3+9509*n^2 +9384*n+3456) * a(n+1) -9*(9*n^2+27*n+20)^2 * a(n) = 0.
G.f.: (1+x)^(-1)*F(1/3,1/3,2/3,2/3;1/2,1/2,1;729*x/16), where F(a1,a2,a3,a4;b1,b2,b3;z) is a hypergeometric series.
a(n) ~ 3^(6*n+7)/(745*Pi*n*2^(4*n+2)). - Vaclav Kotesovec, Aug 06 2013

A188676 Alternate partial sums of the binomial coefficients binomial(3*n,n).

Original entry on oeis.org

1, 2, 13, 71, 424, 2579, 15985, 100295, 635176, 4051649, 25993366, 167543354, 1084134346, 7038291098, 45821937982, 299045487602, 1955803426045, 12815265660680, 84111082917925, 552872886403775, 3638971619401720
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A104859, A188678, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

Programs

  • Mathematica
    Table[Sum[Binomial[3k, k](-1)^(n-k), {k, 0, n}], {n, 0, 20}]
  • Maxima
    makelist(sum(binomial(3*k,k)*(-1)^(n-k),k,0,n),n,0,20);

Formula

a(n) = sum(k=0..n, (-1)^(n-k)*binomial(3*k,k) ).
Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(23*n^2+67*n+48)*a(n+1)-3*(3*n+4)*(3n+5)*a(n)=0.
G.f.: 2*cos((1/3)*arcsin(3*sqrt(3*x)/2))/((1+x)*sqrt(4-27*x)).
a(n) ~ 3^(3*n+7/2)/(62*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

A188681 a(n) = binomial(3*n,n)^2/(2*n+1).

Original entry on oeis.org

1, 3, 45, 1008, 27225, 819819, 26509392, 901402560, 31818681873, 1156122556875, 42985853635725, 1628541825580800, 62667882587091600, 2443473892345873968, 96351855806554401600, 3836565846094702507776, 154071018890153214025473
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188682, A188683, A188684, A188685, A188686, A188687.

Programs

  • Mathematica
    Table[Binomial[3k,k]^2/(2k+1),{k,0,20}]
CoefficientList[Series[HypergeometricPFQ[{1/3,1/3,2/3,2/3}, {1/2,1,3/2}, (729 x)/16],{x,0,20}],x]  (* Harvey P. Dale, Apr 22 2011 *)
  • Maxima
    makelist(binomial(3*k,k)^2/(2*k+1),k,0,20);

Formula

Recurrence: 4*(n+1)^2*(2*n+1)*(2*n+3)*a(n+1)-9*(3*n+1)^2*(3*n+2)^2*a(n)=0.
a(n) ~ 3^(6*n+1)/(Pi*2^(4*n+3)*n^2). - Vaclav Kotesovec, Aug 16 2013

A188679 Partial sums of binomial(3n,n)^2.

Original entry on oeis.org

1, 10, 235, 7291, 252316, 9270325, 353892421, 13874930821, 554792522662, 22521121103287, 925224047453512, 38381686035811912, 1605078750713101912, 67578873844051699048, 2861782692234129345448, 121795323921169907086504
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188678, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

Programs

  • Mathematica
    Table[Sum[Binomial[3k,k]^2,{k,0,n}],{n,0,20}]
Accumulate[Table[Binomial[3n,n]^2,{n,0,20}]] (* Harvey P. Dale, Sep 26 2019 *)
  • Maxima
    makelist(sum(binomial(3*k,k)^2,k,0,n),n,0,20);

Formula

a(n) = sum(C(3k,k)^2, k=0..n).
Recurrence: 4*(2*n^2+7*n+6)^2 * a(n+2) -(745*n^4+4486*n^3+10093*n^2 +10056*n+3744) * a(n+1) +9*(9*n^2+27*n+20)^2 * a(n) = 0.
G.f.: (1-x)^(-1)*F(1/3,1/3,2/3,2/3;1/2,1/2,1;729*x/16), where F(a1,a2,a3,a4;b1,b2,b3;z) is a hypergeometric series.
a(n) ~ 3^(6*n+7)/(713*Pi*n*2^(4*n+2)). - Vaclav Kotesovec, Aug 06 2013

A188683 Alternate partial sums of binomial(3n,n)^2/(2n+1).

Original entry on oeis.org

1, 2, 43, 965, 26260, 793559, 25715833, 875686727, 30942995146, 1125179561729, 41860674073996, 1586681151506804, 61081201435584796, 2382392690910289172, 93969463115644112428, 3742596382979058395348
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188678, A188681, A188686, A188687.
Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), this sequence (k=1), A188685 (k=2).
Cf. Partial sums of binomial(3n,n)^2/(2n+1)^k: A188679 (k=0), A188682 (k=1), A188684 (k=2).

Programs

  • Mathematica
    Table[Sum[Binomial[3k,k]^2(-1)^(n-k)/(2k+1),{k,0,n}],{n,0,20}]
  • Maxima
    makelist(sum(binomial(3*k,k)^2*(-1)^(n-k)/(2*k+1),k,0,n),n,0,20);

Formula

a(n) = sum(binomial(3*k,k)^2*(-1)^(n-k)/(2*k+1), k=0..n).
Recurrence: 4*(n+2)^2*(4*n^2+16*n+15) * a(n+2) -(713*n^4+4246*n^3 +9421*n^2 +9224*n+3360) * a(n+1) -9*(9*n^2+27*n+20)^2 * a(n) = 0.
a(n) ~ 3^(6*n+7)/(745*Pi*n^2*2^(4*n+3)). - Vaclav Kotesovec, Aug 06 2013

A188682 Partial sums of binomials bin(3n,n)^2/(2n+1).

Original entry on oeis.org

1, 4, 49, 1057, 28282, 848101, 27357493, 928760053, 32747441926, 1188869998801, 44174723634526, 1672716549215326, 64340599136306926, 2507814491482180894, 98859670298036582494, 3935425516392739090270, 158006444406545953115743
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Links

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188681, A188683, A188684, A188685, A188686, A188687.

Programs

  • Mathematica
    Table[Sum[Binomial[3k,k]^2/(2k+1),{k,0,n}],{n,0,20}]
Accumulate[Table[Binomial[3n,n]^2/(2n+1),{n,0,20}]] (* Harvey P. Dale, Jul 10 2016 *)
  • Maxima
    makelist(sum(binomial(3*k,k)^2/(2*k+1),k,0,n),n,0,20);

Formula

a(n) = sum(bin(3*k,k)^2/(2*k+1),k=0..n).
Recurrence: 4*(n+2)^2*(4*n^2+16*n+15) * a(n+2) -(745*n^4+4502*n^3+10181*n^2+10216*n+3840) * a(n+1) +9*(9*n^2+27*n+20)^2 *a(n) = 0.
a(n) ~ 3^(6*n+7)/(713*Pi*n^2*2^(4*n+3)). - Vaclav Kotesovec, Aug 06 2013
Showing 1-8 of 8 results.

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